A definition exchange energy?

I'm wondering, what is an exact definition of the exchange energy in atomic physics and/or quantum chemistry ?

For the best of my knowledge, the case is quite simple for correlation energy, namely
[itex]E_{corr} = E_{exact} - E_{HF}[/itex], where [itex]E_{exact}[/itex] is the exact solution of the Schrodinger equation and
[itex]E_{HF}=\left(\Psi_{HF},\hat{H}\Psi_{HF}\right)[/itex] is the Hamiltonian expectation value for the (approximate) complete basis set Hartree-Fock wavefunction. In other words, [itex]E_{corr}[/itex] is "everything beyond HF approximation" (in non-relativistic case of course).

I believe that there is some similar definition for also exchange energy ([itex]E_x[/itex]). But what is it?
It is clear that [itex]E_x[/itex] originates from the Pauli exclusion principle, i.e., the wavefunction symmetry.
So, am I right that one can define [itex]E_x[/itex] as something like
[itex]E_x = E_{HF} - E_H[/itex] where
[itex]E_H[/itex] is the variational Schrodinger equation solution with a Hartree product trial wavefunction instead of Slatter-determinant Hartree-Fock one?

In general, the term "exchange energy" is not uniquely defined on its own. What this means is that this is not a property of a given wave function and Hamiltonian. However, within the Hartree-Fock and Kohn-Sham theories clear meanings can be assigned, namely, the energy contributions to the total energy (within that approximation) due to the exchange part of the Fock operator (HF) or due to the exchange functional (KS).

In general, the term "exchange energy" is not uniquely defined on its own. ... However, within the Hartree-Fock and Kohn-Sham theories clear meanings can be assigned, namely, the energy contributions to the total energy (within that approximation) due to the exchange part of the Fock operator (HF) or due to the exchange functional (KS).

Thanks for your reply!
Isn't it a somewhat puzzling way to define exchange 'part' of the energy via the value of the exchange DFT functional ?
Moreover, it seems to me that your answer implies that so many fundamentally important papers aimed at the development
of purely exchange functionals in the framework of DFT (e.g., classical Becke's 1988 paper is cited over 19 000 times nowadays (!)) deal with the physical quantity, which is 'not uniquely defined' actually
This sounds really surprising...

Well, in DFT there is a clear meaning, because it is a mean-field theory[1]. What I'm saying is that if you have a general, non-determinant wave function, I'm not aware of any commonly accepted definition of its "exchange energy".

I don't remember the details of the B88 paper, and I can't access it now to look it up. But it is mainly cited that many times, because B88 is a part of the BLYP and B3LYP functionals. I wouldn't be surprised if 95% of the people citing it have never even considered reading it :).

[1] But even in DFT exchange and correlation are very hard to separate. This is why there are certain popular combinations of exchange and correlation functional: they need to be combined in certain ways to cancel each other's errors. For example, if you take a DFT correlation functional (without its exchange functional) and calculate its correlation energy by plugging in some HF, MCSCF, or CCSD density, you generally get trash. And if you combine random exchange with random correlation functionals, you also often get funny numbers.